HMMT 十一月 2009 · 冲刺赛 · 第 15 题

HMMT November 2009 — Guts Round — Problem 15

[ 8 ] The curves x + y = 36 and y = x − 7 intersect at four points. Find the sum of the squares of the x -coordinates of these points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nd 2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND 1

解析

[ 8 ] The curves x + y = 36 and y = x − 7 intersect at four points. Find the sum of the squares of the x -coordinates of these points. 2 2 Answer: 26 If we use the system of equations to solve for y , we get y + y − 29 = 0 (since x = y +7). 2 The sum of the roots of this equation is − 1. Combine this with x = y + 7 to see that the sum of the square of the possible values of x is 2 · ( − 1 + 7 · 2) = 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nd 2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND 1